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CHE 185 – PROCESS CONTROL AND DYNAMICS

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CHE 185 – PROCESS CONTROL AND DYNAMICS

FREQUENCY RESPONSE ANALYSIS

- Is the response of a process to a sinusoidal input
- Considers the effect of the time scale of the input.
- Important for understanding the propagation of variability through a process.
- Important for terminology of the process control field.
- But it is NOT normally used for tuning or design of industrial controllers.

- Process Exposed to a Sinusoidal Input.

- Key Components – INPUT FREQUENCY, ampLitude ratio, phase angle.

- Effect of Frequency on Ar and Φ
- PEAK TIME DIFFERENCE

- Bode plot of Ar and Φ versus frequency ω

- BASIS OF BODE PLOT IS A MEASURE OF RELATIVE AMPLITUDE AND PHASE LAG BETWEEN A REGULAR (SINUSOIDAL) SET POINT CHANGE AND THE OUTPUT SIGNAL
- THIS TECHNIQUE INDICATES STABILITY OF THE SYSTEM
- THE ANALYSIS IS COMPLETED WITH AN OPEN LOOP

- Ways to Generate Bode Plot INCLUDE:
- Direct excitation of process.
- Combine transfer function of the process with sinusoidal input.
- Substitute s=iwinto Gp(s) and convert into real and imaginary components which yield Ar(w) and f(w).
- SEE APPLICATION METHOD IN EXAMPLES 11.1 AND 11.2

- Apply a pulse test.

- A SINUSOIDAL SETPOINT IS SENT TO THE LOOP
- AFTER THE SYSTEM REACHES STEADY STATE, THERE IS A LAG, CALLED THE PHASE LAG, BETWEEN THE AMPLITUDE PEAK OF THE INLET SIGNAL AND THE AMPLITUDE PEAK OF THE OUTLET SIGNAL
- THE FREQUENCY IS ADJUSTED SO THE PHASE LAG OF THE OUTLET SIGNAL IS 180° BEHIND THE INPUT SIGNAL

- THE RESULTS ARE THEN APPLIED TO A CLOSED LOOP
- THE SETPOINT IS CHANGED TO A CONSTANT VALUE
- SINCE THE ERROR SIGNAL IS 180° OUT OF PHASE AND IS NEGATIVE RELATIVE TO THE INPUT SIGNAL, IT REINFORCES THE PREVIOUS ERROR SIGNAL

- THE AMPLITUDE OF THE ERROR SIGNAL BECOMES THE OTHER FACTOR
- IF THE AMPLITUDE OF THE ERROR SIGNAL TO THE AMPLITUDE OF THE ORIGINAL SINUSOID SETPOINT, CALLED THE AMPLITUDE RATIO, IS LESS THAN ONE THEN THE ERROR WILL DECAY TO ZERO OVER TIME,
- IF THE AMPLITUDE RATIO WAS EQUAL TO ONE, A PERMANENT STANDING WAVE WILL RESULT
- IF THE AMPLITUDE RATIO WAS GREATER THAN ONE, THE ERROR WILL GROW WITHOUT LIMIT.

- BODE’S STABILITY CRITERION SAYS:
- WHEN THE AMPLITUDE RATIO IS LESS THAN ONE, THE SYSTEM IS STABLE
- WHEN THE AMPLITUDE RATIO IS GREATER THAN ONE, THE SYSTEM IS UNSTABLE
- THE AMPLITUDE RATIO IS DEFINED AS:
WHERE ar REFERS TO AMPLITUDE AS SHOWN IN FIGUREs 11.2.1 and 11.2.2 IN THE TEXT AND ωIS THE FREQUENCY OF THE SINUSOID

- BODE PLOTS
- THESE ARE SHOWN FOR FOPDT PROCESSES IN FIGURES 11.2.2 AND 11.3.2
- A SECOND ORDER PLOT IS FIGURE 11.3.3

- GENERAL TECHNIQUE TO PLOT
- Write the transfer function in proper form (unit value for lowest order term in denominator)
- Separate the transfer function into parts based on poles and zeros
- Draw bode diagram for each part
- Sum the parts to get the final plot

- Transfer function in proper form or
- Parts are based on pole at and constant of 3.3
- Pole plot is constant 0 db up to break ω, then drops off
- Constant has value of 10.4 db

- Function with real poles and zeros

- Function with real poles and zeros

- Function with pole at origin

- Function with repeated real poles,negative constant

- Function with complex conjugate poles

- Function with multiple polesatorigin, complexconjugatezeros

- Function with multiple polesatorigin, complexconjugatezeros

- Function with time delay

- Developing a Bode Plot from the Transfer Function

- Derivation for a First Order Process

- Properties of Bode Plots

- Bode Plot of Complex Transfer Functions
- Break transfer function into a product of simple transfer functions.
- Identify Ar(ω) and Φ(ω) of each simple transfer function from Table 11.1.
- Combine to get Ar(ω) and Φ(ω) for complex transfer function according to properties.
- Plot results as a function of ω.

- BODE PLOTS CAN BE PLOTTED FROM TRANSFER FUNCTIONS
- WE CAN SET UP THE TRANSFER FUNCTION: Y(s) – Gp(s)C(s) WHERE C(s) IS THE SINUSOIDAL INPUT
- AN INVERSE LaPLACE TRANSFORM OF THE RESULT THEN PROVIDES A TIME FUNCTION

- TAKING THIS TO A LIMIT TO ELIMINATE TRANSIENTS THAT WILL DECAY LEAVES THE STANDING WAVE FUNCTION
- THIS CAN BE USED TO EVALUATE Ar AS A FUNCTION OF ω
- AND φ AS A FUNCTION OF ω
- TABLE 11.1 PROVIDES FUNCTIONS TO CALCULATE Ar AND φ FOR A NUMBER OF COMMON TRANSFER FUNCTIONS

- GAIN MARGIN AND PHASE MARGIN
- THE BODE STABILITY CRITERION IS EVALUATED AT THE POINT WHERE φ IS EQUAL TO -180°.
- THE FREQUENCY AT THIS POINT IS CALLED THE CRITICAL FREQUENCY

- GAIN MARGIN AND PHASE MARGIN
- THE VALUE OF Ar CALCULATED AT THE CRITICAL FREQUENCY, Ar* DETERMINES THE PROCESS STABILITY
- THIS IS EXPRESSED AS THE GAIN MARGIN:
- WHEN GM > 1, THE SYSTEM IS STABLE

- GAIN MARGIN AND PHASE MARGIN
- THE PHASE MARGIN IS THE VALUE OF THE PHASE ANGLE AT THE POINT WHERE Ar = 1 AND IS RELATIVE TO THE PHASE ANGLE OF -180°:
(EQUATION 11.3.2)

- THE FREQUENCY WHERE THIS CONDITION OCCURS IS CALLED THE CROSSOVER FREQUENCY

- THE PHASE MARGIN IS THE VALUE OF THE PHASE ANGLE AT THE POINT WHERE Ar = 1 AND IS RELATIVE TO THE PHASE ANGLE OF -180°:

- PULSE TEST
- THIS IS AN OPEN LOOP TEST USED TO OBTAIN THE VALUES NECESSARY TO CREATE A BODE PLOT
- RESULTS COMPARE THE AMPLITUDE AND THE DURATION TIMES FOR THE INPUT AND OUTPUT VALUES FOR AN OPEN LOOP.
- THESE ARE USED WITH EQUATIONS 11.4.1 THROUGH 11.4.8 TO OBTAIN THE BODE PLOT

- PULSE TEST EXAMPLE

- Developing a Pulse Test Process Transfer Function

- Limitations of Transfer Functions Developed from Pulse Tests
- They require an open loop time constant to complete.
- Disturbances can corrupt the results.
- Bode plots developed from pulse tests tend to be noisy near the crossover frequency which affects GM and PM calculations.

- PULSE TEST
- COMBINES THE VALUE OF Ar AND φ ON A SINGLE DIAGRAM
- OTHERWISE IT HAS NO ADVANTAGE OVER THE BODE PLOTS

- REFERENCE FIGURE 11.6.1

- REFERENCE FIGURE 11.6.2

- REFERENCE FIGURE 11.6.2
- At low frequencies, the controller has time to reject the disturbances, i.e., Ar is small.
- At high frequencies, the process filters (averages) out the variations and Ar is small.
- At intermediate frequencies, the controlled system is most sensitive to disturbances.
- The peak frequency indicates the frequency for which a controller is most sensitive.